ISBN-10:
038787707X
ISBN-13:
9780387877075
Pub. Date:
07/08/2010
Publisher:
Springer New York
Mathematical Foundations of Neuroscience / Edition 1

Mathematical Foundations of Neuroscience / Edition 1

by G. Bard Ermentrout, David H. TermanG. Bard Ermentrout
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Overview

This book applies methods from nonlinear dynamics to problems in neuroscience. It uses modern mathematical approaches to understand patterns of neuronal activity seen in experiments and models of neuronal behavior. The intended audience is researchers interested in applying mathematics to important problems in neuroscience, and neuroscientists who would like to understand how to create models, as well as the mathematical and computational methods for analyzing them. The authors take a very broad approach and use many different methods to solve and understand complex models of neurons and circuits. They explain and combine numerical, analytical, dynamical systems and perturbation methods to produce a modern approach to the types of model equations that arise in neuroscience. There are extensive chapters on the role of noise, multiple time scales and spatial interactions in generating complex activity patterns found in experiments. The early chapters require little more than basic calculus and some elementary differential equations and can form the core of a computational neuroscience course. Later chapters can be used as a basis for a graduate class and as a source for current research in mathematical neuroscience. The book contains a large number of illustrations, chapter summaries and hundreds of exercises which are motivated by issues that arise in biology, and involve both computation and analysis. Bard Ermentrout is Professor of Computational Biology and Professor of Mathematics at the University of Pittsburgh. David Terman is Professor of Mathematics at the Ohio State University.

Product Details

ISBN-13: 9780387877075
Publisher: Springer New York
Publication date: 07/08/2010
Series: Interdisciplinary Applied Mathematics , #35
Edition description: 2010
Pages: 422
Product dimensions: 6.10(w) x 9.20(h) x 1.10(d)

Table of Contents

1 The Hodgkin-Huxley Equations 1

1.1 The Resting Potential 1

1.2 The Nernst Equation 3

1.3 The Goldman-Hodgkin-Katz Equation 5

1.4 Equivalent Circuits: The Electrical Analogue 8

1.5 The Membrane Time Constant 11

1.6 The Cable Equation 13

1.7 The Squid Action Potential 16

1.8 Voltage-Gated Channels 18

1.9 Hodgkin-Huxley Model 20

1.10 The Action Potential Revisited 25

1.11 Bibliography 27

1.12 Exercises 28

2 Dendrites 29

2.1 Multiple Compartments 29

2.2 The Cable Equation 33

2.3 The Infinite Cable 34

2.4 Finite and Semi-infinite Cables 36

2.5 Branching and Equivalent Cylinders 38

2.6 An Isolated Junction 40

2.7 Dendrites with Active Processes 42

2.8 Concluding Remarks 45

2.9 Bibliography 45

2.10 Exercises 45

3 Dynamics 49

3.1 Introduction to Dynamical Systems 49

3.2 The Morris-Lecar Model 49

3.3 The Phase Plane 51

3.3.1 Stability of Fixed Points 52

3.3.2 Excitable Systems 53

3.3.3 Oscillations 55

3.4 Bifurcation Analysis 56

3.4.1 The Hopf Bifurcation 56

3.4.2 Saddle-Node on a Limit Cycle 58

3.4.3 Saddle-Homoclinic Bifurcation 60

3.4.4 Class I and Class II 62

3.5 Bifurcation Analysis of the Hodgkin-Huxley Equations 63

3.6 Reduction of the Hodgkin-Huxley Model to a Two-Variable Model 66

3.7 FitzHugh-Nagumo Equations 69

3.8 Bibliography 70

3.9 Exercises 70

4 The Variety of Channels 77

4.1 Overview 77

4.2 Sodium Channels 78

4.3 Calcium Channels 80

4.4 Voltage-Gated Potassium Channels 82

4.4.1 A-Current 83

4.4.2 M-Current 85

4.4.3 The Inward Rectifier 86

4.5 Sag 87

4.6 Currents and Ionic Concentrations 88

4.7 Calcium-Dependent Channels 90

4.7.1 Calcium Dependent Potassium: The Afterhyperpolarization 90

4.7.2 Calcium-Activated Nonspecific Cation Current 93

4.8 Bibliography 95

4.9 Exercises 95

4.10 Projects 100

5 Bursting Oscillations 103

5.1 Introduction to Bursting 103

5.2 Square-Wave Bursters 105

5.3 Elliptic Bursting 111

5.4 Parabolic Bursting 114

5.5 Classification of Bursters 117

5.6 Chaotic Dynamics 118

5.6.1 Chaos in Square-Wave Bursting Models 118

5.6.2 Symbolic Dynamics 121

5.6.3 Bistability and the Blue-Sky Catastrophe 123

5.7 Bibliography 125

5.8 Exercises 126

6 Propagating Action Potentials 129

6.1 Traveling Waves and Homoclinic Orbits 130

6.2 Scalar Bistable Equations 132

6.2.1 Numerical Shooting 135

6.3 Singular Construction of Waves 136

6.3.1 Wave Trains 139

6.4 Dispersion Relations 139

6.4.1 Dispersion Kinematics 141

6.5 Morris-Lecar Revisited and Shilnikov Dynamics 141

6.5.1 Class II Dynamics 142

6.5.2 Class I Dynamics 143

6.6 Stability of the Wave 145

6.6.1 Linearization 146

6.6.2 The Evans Function 147

6.7 Myelinated Axons and Discrete Diffusion 149

6.8 Bibliography 151

6.9 Exercises 152

7 Synaptic Channels 157

7.1 Synaptic Dynamics 158

7.1.1 Gluiamate 161

7.1.2 γ-Aminobutyric Acid 162

7.1.3 Gap or Electrical Junctions 164

7.2 Short-Term Plasticity 164

7.2.1 Other Models 167

7.3 Long-Term Plasticity 168

7.4 Bibliography 169

7.5 Exercises 169

8 Neural Oscillators: Weak Coupling 171

8.1 Neural Oscillators, Phase, and Isochrons 172

8.1.1 Phase Resetting and Adjoints 174

8.1.2 The Adjoint 177

8.1.3 Examples of Adjoints 178

8.1.4 Bifurcations and Adjoints 181

8.1.5 Spike-Time Response Curves 186

8.2 Who Cares About Adjoints? 187

8.2.1 Relationship of the Adjoint and the Response to Inputs 187

8.2.2 Forced Oscillators 189

8.2.3 Coupled Oscillators 193

8.2.4 Other Map Models 199

8.3 Weak Coupling 202

8.3.1 Geometric Idea 203

8.3.2 Applications of Weak Coupling 205

8.3.3 Synaptic Coupling near Bifurcations 206

8.3.4 Small Central Pattern Generators 208

8.3.5 Linear Arrays of Cells 213

8.3.6 Two-Dimensional Arrays 217

8.3.7 All-to-All Coupling 219

8.4 Pulse-Coupled Networks: Solitary Waves 223

8.4.1 Integrate-and-Fire Model 226

8.4.2 Stability 229

8.5 Bibliography 229

8.6 Exercises 229

8.7 Projects 238

9 Neuronal Networks: Fast/Slow Analysis 241

9.1 Introduction 241

9.2 Mathematical Models for Neuronal Networks 242

9.2.1 Individual Cells 242

9.2.2 Synaptic Connections 243

9.2.3 Network Architecture 245

9.3 Examples of Firing Patterns 246

9.4 Singular Construction of the Action Potential 249

9.5 Synchrony with Excitatory Synapses 254

9.6 Postinhibitory Rebound 258

9.6.1 Two Mutually Coupled Cells 258

9.6.2 Clustering 260

9.6.3 Dynamic Clustering 260

9.7 Antiphase Oscillations with Excitatory Synapses 262

9.7.1 Existence of Antiphase Oscillations 263

9.7.2 Stability of Antiphase Oscillations 266

9.8 Almost-Synchronous Solutions 269

9.8.1 Almost Synchrony with Inhibitory Synapses 269

9.8.2 Almost Synchrony with Excitatory Synapses 271

9.8.3 Synchrony with Inhibitory Synapses 274

9.9 Slow Inhibitory Synapses 275

9.9.1 Fast/Slow Decomposition 275

9.9.2 Antiphase Solution 276

9.9.3 Suppressed Solutions 278

9.10 Propagating Waves 278

9.11 Bibliography 282

9.12 Exercises 282

10 Noise 285

10.1 Stochastic Differential Equations 287

10.1.1 The Wiener Process 288

10.1.2 Stochastic Integrals 289

10.1.3 Change of Variables: Itô's Formula 289

10.1.4 Fokker-Planck Equation: General Considerations 290

10.1.5 Scalar with Constant Noise 293

10.1.6 First Passage Times 295

10.2 Firing Rates of Scalar Neuron Models 299

10.2.1 The Fokker-Planck Equation 299

10.2.2 First Passage Times 303

10.2.3 Interspike Intervals 306

10.2.4 Colored Noise 307

10.2.5 Nonconstant Inputs and Filtering Properties 309

10.3 Weak Noise and Moment Expansions 310

10.4 Poisson Processes 314

10.4.1 Basic Statistics 314

10.4.2 Channel Simulations 317

10.4.3 Stochastic Spike Models: Beyond Poisson 319

10.5 Bibliography 321

10.6 Exercises 321

10.7 Projects 326

11 Firing Rate Models 331

11.1 A Number of Derivations 332

11.1.1 Heuristic Derivation 332

11.1.2 Derivation from Averaging 336

11.1.3 Populations of Neurons 338

11.2 Population Density Methods 341

11.3 The Wilson-Cowan Equations 344

11.3.1 Scalar Recurrent Model 346

11.3.2 Two-Population Networks 346

11.3.3 Excitatory-Inhibitory Pairs 350

11.3.4 Generalizations of Firing Rate Models 356

11.3.5 Beyond Mean Field 359

11.4 Some Methods for Delay Equations 361

11.5 Exercises 363

11.6 Projects 365

12 Spatially Distributed Networks 369

12.1 Introduction 369

12.2 Unstructured Networks 370

12.2.1 McCulloch-Pitts 370

12.2.2 Hopfield's Model 371

12.2.3 Designing Memories 373

12.3 Waves 375

12.3.1 Wavefronts 376

12.3.2 Pulses 379

12.4 Bumps 383

12.4.1 The Wilson-Cowan Equations 384

12.4.2 Stability 387

12.4.3 More General Stability 388

12.4.4 More General Firing Rates 389

12.4.5 Applications of Bumps 390

12.5 Spatial Patterns: Hallucinations 394

12.6 Exercises 399

References 407

Index 419

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